Patrick Suppes

Contemporary Developments in Mathematical Psychology

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Mathematical Methods in the Social Sciences

MMSS is an adjunct major and must be completed with a standalone major in a social science or other approved area. See the Mathematics Second Major or Minor for MMSS Students (https:// catalogs.northwestern.edu/undergraduate/arts-sciences/mathematics/ mathematics-second-major-minor-mmss-students) for information about the major or the minor in mathematics when combined with the MMSS adjunct major; see the program website (https:// www.mmss.northwestern.edu/undergraduate) and relevant sections of this Catalog for information on adjustments to requirements in other majors for students in MMSS.

Foundational Aspects of Theories of Measurement

It is a scientific platitude that there can be neither precise control nor prediction of phenomena without measurement. Disciplines as diverse as cosmology and social psychology provide evidence that it is nearly useless to have an exactly formulated quantitative theory, if empirically feasible methods of measurement cannot be developed for a substantial portion of the quantitative concepts of the theory. Given a physical concept like that of mass or a psychological concept like that of habit strength, the point of a theory of measurement is to lay bare the structure of a collection of empirical relations which may be used to measure the characteristic of empirical phenomena-corresponding to the concept. Why a collection of relations? From an abstract standpoint, a set of empirical data consists of a collection of relations between specified objects. For example, data on the relative weights of a set of physical objects are easily represented by an ordering relation on the set; additional data, and a fortiori an additional relation, are needed to yield a satisfactory quantitative measurement of the masses of the objects.

Models of Data

To nearly all the members of this Congress, the logical notion of a model of a theory is too familiar to need detailed review here. Roughly speaking, a model of a theory may be defined as a possible realization in which all valid sentences of the theory are satisfied, and a possible realization of the theory is an entity of the appropriate set-theoretical structure. For instance, we may characterize a possible realization of the mathematical theory of groups as an ordered couple whose first member is a nonempty set and whose second member is a binary operation on this set. A possible realization of the theory of groups is a model of the theory if the axioms of the theory are satisfied in the realization, for in this case (as well as in many others), the valid sentences of the theory are defined as those sentences which are logical consequences of the axioms. To provide complete mathematical flexibility I shall speak of theories axiomatized within general set theory by defining an appropriate set-theoretical predicate (e.g., ‘is a group’) rather than of theories axiomatized directly within first-order logic as a formal language. For the purposes of this paper, this difference is not critical. In the set-theoretical case, it is convenient sometimes to speak of the appropriate predicate’s being satisfied by a possible realization. But whichever sense of formalization is used, essentially the same logical notion of model applies.1

When are probabilistic explanations possible

The primary criterion of adequacy of a probabilistic causal analysis is that the causal variable should render the simultaneous phenomenological data conditionally independent. The intuition back of this idea is that the common cause of the phenomena should factor out the observed correlations. So we label the principle the common cause criterion. If we find that the barometric pressure and temperature are both dropping at the same time, we do not think of one as the cause of the other but look for a common dynamical cause within the physical theory of meteorology. If we find fever and headaches positively correlated, we look for a common disease as the source and do not consider one the cause of the other. But we do not want to suggest that satisfaction of this criterion is the end of the search for causes or probabilistic explanations. It does represent a significant and important milestone in any particular investigation.

OUTLINES OF A FORMAL THEORY OF VALUE, I

Physics has advanced, however, without answering metaphysical questions; statistics has advanced without answering semantical questions; and it is our opinion that similar substantial progress in the theory of value can be made independent of metaphysics and semantics. As in other disciplines, theory in the domain of value can proceed along formal lines without waiting upon a solution to the grand questions; indeed even the most modest constructive progress might result, here as elsewhere, in putting what have been considered the fundamental problems in a new light. We take it as the general function of formal value theory to provide formal criteria for rational decision, choice and evaluation. Our conception of this aspect of value theory is in one way similar to Kants, for like him we believe it possible to state in purely formal terms certain necessary conditions for rationality with respect to value. Unlike Kant, however, we do not suggest that any particular evaluations or value principles can be derived from purely formal considerations. Value theory, as here conceived, is associated with another venerable, and at present rather unfashionable, tradition, for it seems to us that there is a sense in which it is perfectly correct to say that just as logic can be used to define neces1 The unexpected death of Professor J. C. C. McKinsey after the completion of an earlier and much shorter draft of the present paper means that although he played a major part in formulating the fundamental ideas he cannot be held accountable for any of the shortcomings of the final version.

Information and inference

I. Information and Induction.- On Semantic Information.- Bayesian Information Usage.- Experimentation as Communication with Nature.- II. Information and Some Problems of the Scientific Method.- On the Information Provided by Observations.- Quantitative Tools for Evaluating Scientific Systematizations.- Qualitative Information and Entropy Structures.- III. Information and Learning.- Learning and the Structure of Information.- IV. New Applications of Information Concepts.- Surface Information and Depth Information.- Towards a General Theory of Auxiliary Concepts and Definability in First-Order Theories.- Index of Names.- Index of Subjects.

Stimulus-response theory of finite automata

Abstract The central aim of the paper is to state and prove a representation theorem for finite automata in terms of models of stimulus-response theory. The main theorem is that, given any connected finite automaton, there is a stimulus-response model that asymptotically becomes isomorphic to it. Implications of this result for language learning are discussed in some detail. In addition, an immediate corollary is that any tote hierarchy in the sense of Miller and Chomsky is isomorphic to some stimulus-response model at asymptote. Representations of probabilistic automata are also discussed, and an application to the learning of arithmetic algorithms is given.

The Place of Theory in Educational Research

very much aware that educational research is a minor activity compared with education as a whole. All of us probably feel on occasion that there is little hope that educational research, given the small national effort devoted to it, will have any real impact on education as a whole. Such pessimistic thoughts are not historically, I think, supported by the evidence, especially when we look at the evidence outside of education as

A Set of Independent Axioms for Extensive Quantities

The modern viewpoint on quantities goes back at least to Newton’s Universal Arithmetick. Newton asserts that the relation between any two quantities of the same kind can be expressed by a real, positive number.2 In 1901, O. Hoelder gave a set of ‘Axiome der Quantitaet’, which are sufficient to establish an isomorphism between any realization of his axioms and the additive semigroup of all positive real numbers. Related work of Hilbert, Veronese and others is indicative of a general interest in the subject of quantities in the abstract on the part of mathematicians of this period. During the last thirty years, from another direction, philosophers of science have become interested in the logical analysis of empirical procedures of measurement.3 The interests of these two groups overlap insofar as the philosophers have been concerned to state the formal conditions which must be satisfied by empirical operations measuring some characteristic of physical objects (or other entities). Philosophers have divided quantities (that is, entities or objects considered relatively to a given characteristic, such as mass, length or hardness) into two kinds. Intensive quantities are those which can merely be arranged in a serial order; extensive quantities are those for which a “natural” operation of addition or combination can also be specified. Another, more exact, way of making a distinction of this order is to say that intensive quantities are quantities to which numbers can be assigned uniquely up to a monotone transformation, and extensive quantities are quantities to which numbers can be assigned uniquely up to a similarity transformation (that is, multiplication by a positive constant).4 This last condition may be said to be the criterion of formal adequacy for a system of extensive quantities.